Holography for six dimensional theories: a universal framework

It is challenging to define higher dimensional quantum field theories. For example, a Lagrangian description is often lacking. Moreover, gauge theories are non renormalizable in d > 4. A possible ultraviolet completion is string theory, which is a natural arena for the study of many interesting field theories. In particular, string theory provides us with a powerful tool for the study of conformal field theories: the AdS/CFT correspondence. The AdS/CFT is a duality that relates a theory of quantum gravity in Anti de Sitter space in (p+2)-dimensions to a conformal field theory in (p+1)-dimensions, an explicit realization of the so called holographic principle which is possible thanks to the presence of Dp-branes. These are p-dimensional objects that act as sources of gravity, carrying a (p+1)-dimensional gauge theory on their world volume. In the so called decoupling limit, the gauge degrees of freedom living on the brane decouple from gravity, which results in a AdS(p+2) vacuum state. In this thesis we use holography to study six dimensional theories. In particular we consider the so called (1, 0) conformal field theories, which are the half supersymmetric cousins of the more popular (2,0) theories living on M5 branes, whose gravity dual is the maximally supersymmetric vacuum AdS7 × S4 of M-theory. The simplest example of (1, 0) theory can be obtained orbifolding this solution. More general (1, 0) theories can be constructed in the AdS/CFT setup via brane configurations involving NS5, D6 and D8 branes. In the decoupling limit these configurations give rise to AdS7 vacua in massive type IIA supergravity. We give a complete classification of these vacua, with an infinite class of analytic solutions that correspond to infinite possible configurations of branes. The resulting internal space M3 has the topology of a three sphere, with SU(2) isometry and physical singularities at the points where the branes are located. We also classify all possible compactifications of these solutions, obtained wrapping the branes on two, three and four-manifolds of negative curvature, leading to AdS5, AdS4 and AdS3 vacua. Complete flows connecting AdS7 to the lower dimensional vacua can be constructed, suggesting the existence of a renormalization group flow between the six dimensional CFT at high energies and a lower dimensional CFT in four, three, two-dimensions at low energies. Even though these lower dimensional theories are not yet known, they now have a holographic description. The AdS4 solutions are also interesting as four-dimensional vacua with localized sources. The compactification procedure is so universal it suggests to consider a complete reduction ansatz of type IIA supergravity on the internal space M3. Indeed, the dimensional reduction leads to minimal gauged supergravity in seven dimensions, a theory with sixteen supercharges and an SU(2) gauge field. This reduction procedure does not depend on the details of the internal space M3 and thereby on the particular choice of brane configuration. We can thus claim to have a universal consistent truncation for 6d/7d gauge/gravity duals.